DP IB Physics: SL

B. The particulate nature of matter

B.4 Thermodynamic

DP IB Physics: SL B. The particulate nature of matter B.4 Thermodynamic Understandings Students should understand:
a)	That the first law of thermodynamics as given by [math]Q = ΔU + W[/math] results from the application of conservation of energy to a closed system and relates the internal energy of a system to the transfer of energy as heat and as work
b)	That the work done by or on a closed system as given by [math]W = PΔV[/math] when its boundaries are changed can be described in terms of pressure and changes of volume of the system
c)	That the change in internal energy as given by [math]\Delta U = \frac{3}{2} N k_B \Delta T = \frac{3}{2} n R \Delta T[/math] of a system is related to the change of its temperature
d)	That entropy S is a thermodynamic quantity that relates to the degree of disorder of the particles in a system
e)	That entropy can be determined in terms of macroscopic quantities such as thermal energy and temperature as given by [math]\Delta S = \frac{\Delta Q}{T}[/math] and also in terms of the properties of individual particles of the system as given by [math]S = k_B ln Ω[/math] where [math]k_B[/math] is the Boltzmann constant and Ω is the number of possible microstates of the system
f)	That the second law of thermodynamics refers to the change in entropy of an isolated system and sets constraints on possible physical processes and on the overall evolution of the system
g)	That processes in real isolated systems are almost always irreversible and consequently the entropy of a real isolated system always increases
h)	That the entropy of a non-isolated system can decrease locally, but this is compensated by an equal or greater increase of the entropy of the surroundings
i)	That isovolumetric, isobaric, isothermal and adiabatic processes are obtained by keeping one variable fixed
j)	That adiabatic processes in monatomic ideal gases can be modelled by the equation as given by [math]P V^{5/3} = \text{constant}[/math]
k)	That cyclic gas processes are used to run heat engines
l)	That a heat engine can respond to different cycles and is characterized by its efficiency as given by [math]\eta = \frac{\text{useful work}}{\text{input energy}}[/math]
m)	That the Carnot cycle sets a limit for the efficiency of a heat engine at the temperatures of its heat reservoirs as given by [math]\eta_{\text{Carnot}} = 1 – \frac{T_c}{T_h}[/math]

a) The first law of thermodynamics and related equations:
The first law of thermodynamics is a statement of energy conservation applied to thermodynamic systems. It describes how heat (Q) added to a system result in changes in internal energy (ΔU) and work (W) done by or on the system.
[math]Q = ΔU + W[/math]
Where:
– Q = heat energy transferred to the system (J),
– ΔU = change in internal energy of the system (J),
– W = work done by the system on its surroundings (J).
This equation expresses that the total energy input into a closed system is either stored as internal energy or used to do work on the surroundings.
Figure 1 First law of thermodynamics
b) Understanding the First Law of Thermodynamics
⇒ Internal Energy (U)
Internal energy is the total kinetic and potential energy of a system’s molecules.
In an ideal gas, internal energy is purely kinetic since intermolecular forces are negligible.
– For a monatomic ideal gas, the internal energy depends only on temperature and is given by:
[math]U = \frac{3}{2} N k_B T = \frac{3}{2} n R T[/math]
Where:
– N = total number of molecules,
– [math]k_B[/math] = Boltzmann’s constant ([math][/math])
– T = temperature (K),
– n = number of moles,
– R = universal gas constant (314 J/mol·K).
⇒ Change in Internal Energy:
[math]ΔU = \frac{3}{2} N k_B ΔT = \frac{3}{2} n R ΔT[/math]
This means that any temperature change results in a change in internal energy.
⇒ Work Done by or on the System (W)
If a system expands or contracts, it performs work on its surroundings, given by:
[math]W = PΔV[/math]
Where:
– P = pressure (Pa),
– ΔV = change in volume (m³).
– If ΔV > 0 (expansion), the gas does work on its surroundings (energy leaves the system).
– If ΔV < 0 (compression), the surroundings do work on the gas (energy enters the system).
c) Application of the First Law in Different Processes

Thermodynamic Process	Work Done W	Heat Q	Internal Energy ΔU
Isothermal (ΔT=0)	[math]W = Q[/math]	Heat added equals work done	[math]ΔU = 0[/math]
Isobaric (P constant)	[math]W = PΔT[/math]	Heat increases U and does work	[math]ΔU = Q – W[/math]
Isochoric (ΔV=0)	[math]W = 0[/math]	All heat goes into ΔU	[math]ΔU = Q[/math]
Adiabatic (Q=0)	[math]W = -ΔU[/math]	No heat exchange	ΔU changes due to work

⇒ Isothermal Process (ΔT=0)
– Since temperature is constant, internal energy does not change.
– Heat added is fully converted into work.
– First law simplifies to:
[math]Q = W[/math]
⇒ Isobaric Process (P Constant)
Pressure remains constant while volume changes.
Work is given by:
[math]W = PΔU[/math]
The heat energy is split between internal energy change and work done.
⇒ Isochoric Process (ΔV=0)
– Volume remains constant, so no work is done.
– All heat energy increases internal energy:
[math]ΔU = Q[/math]
⇒ Adiabatic Process (Q=0)
– No heat is exchanged with the surroundings.
– The gas expands or compresses only by doing work:
[math]W = -ΔU[/math]
– If the gas expands, ΔU decreases, and the temperature drops.
– If the gas is compressed, ΔU increases, and the temperature rises.
Figure 2 Thermodynamic processes
d) Entropy and the second law of thermodynamics
⇒ Entropy
Entropy (S) is a thermodynamic quantity that measures the degree of disorder or randomness of a system. It describes the number of ways a system can be arranged at the microscopic level while maintaining the same macroscopic properties (e.g., pressure, temperature, and volume).
– A highly ordered system has low entropy (e.g., a solid crystal).
– A highly disordered system has high entropy (e.g., a gas with molecules moving freely).
Entropy is a state function, meaning it depends only on the system’s state and not on the path taken to reach that state.
⇒ Macroscopic Definition of Entropy
Entropy change (ΔS) can be determined using the heat transfer (ΔQ) and the temperature (T) of the system:
[math]ΔS = TΔQ[/math]
Where:
– ΔS = change in entropy (J/K),
– ΔQ = heat energy added or removed (J),
– T = absolute temperature (K).
If heat is added to a system (ΔQ>0), entropy increases.
If heat is removed (ΔQ<0), entropy decreases.
If a process is reversible, entropy change can be calculated precisely using this formula.
If a process is irreversible, the entropy of the system and surroundings always increases.
Figure 3 The second law of thermodynamics
e) Microscopic Definition of Entropy (Boltzmann’s Entropy Equation)
Entropy is also linked to the number of microstates (Ω) of a system, which represents the different ways particles can be arranged while maintaining the same overall macroscopic properties.
Boltzmann’s entropy equation:
[math]S = k_B lnΩ[/math]
Where:
– S = entropy (J/K),
– [math]k_B[/math]= Boltzmann constant (38×10⁻²³ J/K),
– Ω = number of microstates (possible microscopic arrangements of the system).
If a system has more microstates (Ω increases), entropy increases.
If a system becomes more ordered (Ω decreases), entropy decreases.
Perfect order (Ω=1) means zero entropy.
f) The Second Law of Thermodynamics
The Second Law of Thermodynamics states:
– The total entropy of an isolated system always increases over time for any natural (irreversible) process.
⇒ Mathematically:
[math]\Delta S_{\text{total}} ≥ 0[/math]
Where:
[math]\Delta S_{\text{total}} = \Delta S_{\text{system}} + \Delta S_{\text{surroundings}}[/math]
⇒ Implications of the Second Law
1. Natural Processes Increase Entropy
– Heat spontaneously flows from hot to cold (never the reverse without external work).
– Gases expand to fill a volume instead of staying in one corner of a container.
– Ice melts at room temperature rather than forming spontaneously.
2. Reversible vs. Irreversible Processes
– Reversible processes ([math]\Delta S_{\text{total}} = 0[/math]): These are idealized processes that can be undone without increasing entropy (e.g., slow, frictionless changes in a system).
– Irreversible processes ([math]\Delta S_{\text{total}} > 0[/math]): Most real-world processes involve entropy increase (e.g., friction, heat dissipation, mixing of gases).
Figure 4 Entropy and second law of thermodynamics
3. Heat Engines and Efficiency Constraints
– The Carnot Efficiency (η) represents the theoretical maximum efficiency of a heat engine, limited by entropy considerations.
[math]\eta = 1 – \frac{T_C}{T_H}[/math]
Where [math]T_H[/math] is the temperature of the hot reservoir, and [math]T_C[/math] is the temperature of the cold reservoir.
Figure 5 Carnot efficiency
4. Entropy and the Direction of Time
– The increase of entropy provides a “thermodynamic arrow of time”, meaning systems evolve naturally toward greater disorder.
– This is why breaking an egg is easy, but reversing the process (reassembling it) is practically impossible.
g) Entropy and Irreversibility in Real Isolated Systems
⇒ Irreversible Processes and the Increase of Entropy
The Second Law of Thermodynamics states that in a real isolated system, entropy (S) always increases or remains constant but never decreases:
[math]ΔS_{total} ≥ 0[/math]
⇒ Real Processes Irreversible:
In real-world systems, most processes are irreversible due to:
1. Friction and Dissipation – Mechanical energy is lost as heat.
2. Heat Flow – Heat always flows spontaneously from a hot body to a cold body.
3. Mixing of Substances – Once two gases mix, they do not spontaneously separate.
4. Unidirectional Nature of Natural Processes – A shattered glass does not reassemble itself.
Examples of Irreversible Processes
– Ice melting in water.
– Air leaking out of a tire.
– A car braking due to friction.
– A burning log turning into ash.
Thus, in any real isolated system, entropy increases over time, leading to a natural tendency toward disorder.
h) Entropy Changes in Non-Isolated Systems
A non-isolated system can exchange heat and work with its surroundings. Unlike an isolated system, the entropy of a part of a non-isolated system can decrease, but only if the entropy of the surroundings increases by an equal or greater amount.
[math]\Delta S_{\text{system}} + \Delta S_{\text{surrounding}} \geq 0[/math]
⇒ Examples of Local Decreases in Entropy
1. Living Organisms
– A growing tree reduces entropy by forming highly ordered structures (leaves, wood, etc.).
– However, this process requires energy from the Sun, which increases the entropy of the surroundings (e.g., radiation emission from the Sun).
2. Refrigerators and Air Conditioners
– A refrigerator extracts heat from inside (decreasing entropy).
– But it releases more heat to the room, increasing overall entropy.
3. Freezing Water in a Freezer
– Water forms structured ice crystals (decreasing entropy).
– However, the freezer dumps more heat into the surroundings, increasing entropy.
Thus, entropy can locally decrease, but it is always accompanied by a greater increase in the entropy of the environment.
i) Thermodynamic Processes: Isovolumetric, Isobaric, Isothermal, and Adiabatic
Thermodynamic processes are classified based on which variable remains constant:

Process	Variable Kept Constant	Heat Transfer (Q)	Work Done (W)
Isovolumetric (Isochoric)	Volume (V) constant	Q = ΔU, no work is done	[math]W = 0[/math]
Isobaric	Pressure (P) constant	[math]Q = ΔU + W[/math]	[math]W = PΔV[/math]
Isothermal	Temperature (T) constant	Q = W, internal energy does not change	[math]ΔU = 0[/math]
Adiabatic	No heat exchange (Q=0)	[math]W = -ΔU[/math]	No heat transfer

⇒ Isovolumetric (Isochoric) Process:
– Definition: A thermodynamic process in which volume remains constant (ΔV=0).
-Implication: Since volume does not change, no work is done (W=0).
⇒ First Law of Thermodynamics:
[math]Q = ΔU[/math]
Example: Heating a gas in a rigid container.
⇒ Isobaric Process
Definition: A process where pressure remains constant.
Work Done: Since the volume changes, the system does work on or by the surroundings:
[math]W = PΔV[/math]
⇒ First Law of Thermodynamics:
[math]Q = ΔU + W[/math]
Example: Boiling water in an open container (pressure remains atmospheric while volume increases).
⇒ Isothermal Process
Definition: A process in which temperature remains constant (ΔT=0).
Implication: Since internal energy depends on temperature, the internal energy does not change:
[math]ΔU = 0[/math]
First Law of Thermodynamics simplifies to:
[math]Q = W[/math]
– Example: Slow compression or expansion of a gas in thermal contact with a reservoir.
⇒ Adiabatic Process
Definition: A process in which no heat is exchanged with the surroundings (Q=0).
Implication: All energy transfer is in the form of work:
[math]W = -ΔU[/math]
– Example: Rapid compression of a gas (e.g., when pumping air into a bicycle tire, the air gets hot because heat is not transferred away fast enough).
j) Adiabatic Processes in Monatomic Ideal Gases
⇒ An Adiabatic Process
An adiabatic process is a thermodynamic process in which no heat is exchanged between the system and its surroundings:
[math]Q = 0[/math]
Since no heat is transferred, any change in internal energy (ΔU) must be due to work done on or by the system:
[math]ΔU = -W[/math]
Adiabatic Equation for Monatomic Ideal Gases
For a monatomic ideal gas, the adiabatic process follows the equation:
[math]PV^γ = \text{ constant}[/math]
Where:
– P = Pressure
– V = Volume
– γ = Adiabatic Index or Heat Capacity Ratio, given by:
[math]\gamma = \frac{C_p}{C_v}[/math]
For a monatomic gas, [math]\gamma = \frac{5}{3}[/math], leading to:
[math]PV^{5⁄3} = \text{constant}[/math]
This equation describes how pressure and volume change during an adiabatic process.
Figure 6 Adiabatic process
Properties of Adiabatic Processes
1. Rapid Compression or Expansion
– If a gas is compressed quickly, its temperature rises (e.g., diesel engine compression).
– If a gas expands quickly, its temperature drops (e.g., air released from a compressed can).
2. Work Done in an Adiabatic Process
The work done in an adiabatic process can be found using:
[math]W = \frac{P_1 V_1 – P_2 V_2}{\gamma – 1}[/math]
k) Cyclic Gas Processes and Heat Engines
⇒ Heat Engine
A heat engine is a system that converts heat energy into mechanical work. It operates in a cyclic process, meaning the system returns to its initial state after each cycle.
⇒ Basic Components of a Heat Engine
A typical heat engine consists of:
A hot reservoir at temperature T_h that provides input heat energy (Q_h ).
A working substance (e.g., gas) that undergoes cyclic processes.
A cold reservoir at temperature T_c , where heat is rejected (Q_c ).
A mechanical system that extracts useful work (W).
Figure 7 Cyclic processes and heat engines
l) Heat Engine Cycle
The cycle follows three main steps:
1. Absorbing heat (Q_h) from the high-temperature reservoir.
2. Performing useful work (W) by expanding gas.
3. Releasing excess heat (Q_c ) into the low-temperature reservoir.
Since the process is cyclic, by the First Law of Thermodynamics:
[math]Q_h = W + Q_c[/math]
⇒ Efficiency of a Heat Engine
The efficiency η of a heat engine is given by:
[math]\eta = \frac{\text{Useful Work Output}}{\text{Input Heat Energy}}[/math]
Substituting [math]W = Q_h – Q_c[/math]:
[math]\eta = \frac{Q_h – Q_c}{Q_h} = 1 – \frac{Q_c}{Q_h}[/math]
This equation tells us that an engine is more efficient if it minimizes the rejected heat [math]Q_c[/math].
m) The Carnot Cycle and Maximum Efficiency
⇒ Definition of the Carnot Cycle
The Carnot cycle is a theoretical cycle that defines the maximum possible efficiency of a heat engine operating between two temperatures.
⇒ Carnot Cycle Stages
The cycle consists of four reversible processes:
1. Isothermal Expansion – The gas absorbs heat from the hot reservoir and expands, doing work.
2. Adiabatic Expansion – The gas expands without heat exchange, causing it to cool down.
3. Isothermal Compression – The gas releases heat to the cold reservoir as it compresses.
4. Adiabatic Compression – The gas compresses without heat exchange, causing it to heat up.
⇒ Carnot Efficiency Equation
The maximum possible efficiency of a heat engine operating between two temperature reservoirs ([math]T_h[/math] and [math]T_c[/math]) is given by:
[math]\eta_{\text{Carnot}} = 1 – \frac{T_C}{T_h}[/math]
Where:
[math]T_h[/math] = Temperature of the hot reservoir (in Kelvin
[math]T_c[/math] = Temperature of the cold reservoir (in Kelvin)
⇒ Implications of the Carnot Cycle
1. The efficiency increases as [math]T_c[/math] decreases
– If [math]T_c → 0K[/math], the efficiency approaches 100%, but absolute zero is impossible to reach.
2. No real engine can exceed the Carnot efficiency
– The Carnot cycle is an idealized model with no friction or energy dissipation.

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DP IB Physics: SL

B. The particulate nature of matter

B.4 Thermodynamic

DP IB Physics: SL

B. The particulate nature of matter

B.4 Thermodynamic

a) The first law of thermodynamics and related equations:

b) Understanding the First Law of Thermodynamics

c) Application of the First Law in Different Processes

d) Entropy and the second law of thermodynamics

e) Microscopic Definition of Entropy (Boltzmann’s Entropy Equation)

f) The Second Law of Thermodynamics

g) Entropy and Irreversibility in Real Isolated Systems

h) Entropy Changes in Non-Isolated Systems

i) Thermodynamic Processes: Isovolumetric, Isobaric, Isothermal, and Adiabatic

j) Adiabatic Processes in Monatomic Ideal Gases

k) Cyclic Gas Processes and Heat Engines

l) Heat Engine Cycle

m) The Carnot Cycle and Maximum Efficiency

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